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Constructive Wall-Crossing

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Constructive Wall-Crossing Constructive Wall-Crossing PiljinYi (KIAS) Yongpyung 2012 1102.1729 with Sungjay Lee 1107. 0723 with Heeyeon Kim, Jaemo Park, ZhaolongWang wall-crossing is supersymmetry breaking supersymmetry is a square-root of translational symmetry supercharges are square-root of Hamiltonian wall-crossing is supersymmetry breaking supersymmetry breaking = positive energy vacuum wall-crossing is supersymmetry breaking prototype : (anomalous) N=1 SQED integrate out auxiliary field D wall-crossing is supersymmetry breaking prototype : (anomalous) N=1 SQED wall-crossing is supersymmetry breaking with positive charges - sidesusy preserved susy broken + side : a wall that separates susy region from susy broken region wall-crossing is supersymmetry breaking existence of such walls is a generic feature of systems with 4 or less supercharges susy broken susy preserved : a wall that separates susy region from susy broken region wall-crossing occurs in • D=4 systems: N=1 supersymmetric field theories with anomalous U(1) type IIB string theory on Calabi-Yau with 3-form fluxes type IIA string theory on Calabi-Yau + half-wrapped D6 Heterotic/type I on Calabi-Yau with gauge bundles ……. in other words : entire String Theory Landscape of phenomenological relevance for our universe wall-crossing of BPS particles • D=1 systems (namely, particle-like states): supersymmetric kinks in D=2 N=(2,2) theories supersymmetric (BPS) states in D=4 N=2 Seiberg-Witten theory supersymmetric (BPS) black holes in D=4 N=2 supergravity completely wrapped D-branes in type II string theories on Calabi-Yau canonical example : D=4 N=2 Seiberg-Witten SU(2) U(1) 1996 Bilal + Ferarri self-consistent minimal set of BPS states to SU(2) Seiberg-Witten via monodromy a conjecture for the complete BPS spectrum why does the wall-crossing happen and is the conjecture true ? in other D=4 N=2 Seiberg-Witten theories and supergravities ? ? 1998 Lee + P.Y. N=4 SU(n) ¼ BPS states via multi-center classical dyon solitons 1999 Bak + Lee + Lee + P.Y. N=4 SU(n) ¼ BPS states via multi-center monopole dynamics 1999-2000 Gauntlett + Kim + Park + P.Y. / Gauntlett + Kim + Lee + P.Y. / Stern + P.Y. N=2 SU(n) BPS states via multi-center monopole dynamics 2001 Denef: N=2 supergravity via classical multi-center black holes attractor solutions a generic BPS state is a loose bound state of many charge centers outside particles are loose bound states of inside particles wall-crossing = dissociation of supersymmetric bound states wall-crossing = disappearance of supersymmetry particle states bound states of many charge centers become infinitely large therefore, if viewed from the opposite side wall-crossing backward appearance of supersymmetric multi-center bound states how to count & classify such supersymmetric bound states 2000 Stern + P.Y. general wall-crossing formula for simple total magnetic charges; weak coupling regime 2002 Denef: quiver dynamics representation of N=2 supergravity BH’s ………… 2008 de Boer + El Showk + Messamah + van den Bleeken 2- & 3- particle bound state conjecture 2010/2011 Manschot + Pioline + Sen: general n-particle conjecture 2011 S. Lee + P.Y. / H. Kim + J. Park + Z. Wang + P.Y. reformulation and full solution via generic dyon/BH dynamics constructive and first-principle derivation of complete wall-crossing formula from Seiberg-Witten theory; origin of rational invariants explained from dynamics main result (i) : primary index formula for generic n-body problems Kim+Park+P.Y.+Wang 2011 Manschot, Pioline, Sen 2010/2011 main result (ii) : bound state counting with Bose/Fermi statistics incorporated Kim+Park+P.Y.+Wang 2011 Manschot, Pioline, Sen 2010/2011 from which we find, for example, and which results in BPS states for extended superalgebra in D=4 field theory under the angular momentum/Lorentz boost, transform as a vector and a Dirac spinor, respectively BPS multiplet in rest frame BPS multiplet in rest frame BPS multiplet in rest frame BPS multiplet in rest frame [ a spin ½ + two spin 0 ] x [ angular momentum multiplet ] 2nd helicity trace as the BPS state counter [ a spin ½ + two spin 0 ] x [ angular momentum multiplet ] 2nd helicity trace as the BPS state counter [ a spin ½ + two spin 0 ] x [ angular momentum multiplet ] non-BPS multiplet in rest frame [ a spin 1 + two spin ½ + five spin 0 ] x [ angular momentum multiplet ] more generally, we wish to compute the protected spin character D=4 N=2 superymmetric field theory the smallest multiplet with massless vector: Coulomb phase all off-diagonal fields are massive and can be integrated out D=4 N=2 superymmetric field theory for SU(r+1) D=4 N=2 superymmetric field theory related by extended supersymmetry D=4 N=2 superymmetric field theory SU(2) U(1) case where off-diagonal particles of mass is already integrated out SU(2) U(1) case SU(2) U(1) case BPS particles: electrically / magnetically charged particles with partially preserved susy D=4 N=2 central charge with electric charges and magnetic charges D=4 N=2 central charge with electric charges and magnetic charges let us construct general BPS bound states as a preliminary exercise, consider one dynamical dyon at a time a probe charge to a system of background “core” dyons background dyons: one probe dyon: one dynamical charge in the background dyons a probe charge to a system of background “core” dyons a probe charge to a system of background “core” dyons take one last step: what happens when the imaginary part is very small ? wall-crossing in the core-probe approximation mini wall-crossing problem similar to Schroedinger atom problems +side - side wall-crossing in the core-probe approximation one must elevate this to a N=4 supersymmetric quantum mechanics N=4 SUSY QM with 3 bosons & 4 fermions ? the easiest way to understand this set of multiplets is as a dimensional reduction of on-shell N=1 D=4 vector multiplet (in the Wess-Zumino gauge) N=4 SUSY QM with 3 bosons & 4 fermions ? the easiest way to understand this set of multiplets is as a dimensional reduction of on-shell N=1 D=4 vector multiplet (in the Wess-Zumino gauge) Smilga; Ivanov; Papadopoulos; …… circa1988-1991 N=4 SUSY QM with 3 bosons & 4 fermions full wall-crossing problem : counting BPS bound states of an arbitrary collection of dyons +side - side N=4 SUSY QM with 3n bosons & 4n fermions ? position of A-th dyon N=4 SUSY QM with 3n bosons & 4n fermions ? position of A-th dyon N=4 SUSY QM with 3n bosons & 4n fermions ? position of A-th dyon N=4 SUSY QM with 3n bosons & 4n fermions ? position of A-th dyon N=4 SUSY QM with 3n bosons & 4n fermions ? is N=4 supersymmetric if and only if N=4 SUSY QM with 3n bosons & 4n fermions ? is N=4 supersymmetric with N=4 quantum mechanics for aribtrary collections of dyons Kim+Park+P.Y.+Wang 2011 counting BPS Bound States with N=4 SUSY QM counting BPS bound states with N=4 SUSY QM supersymmetry zero energy wavefunction supported on submanifold ? counting BPS bound states with N=4 SUSY QM supersymmetric lowest landau level problem on submanifold ? wall-crossing problem bound state counting Witten index for this, only one of the four supersymmetries is needed we are allowed to deform the dynamics as long as this preserves one supersymmetry and does not alter the asymptotics dynamics wall-crossing problem bound state counting Witten index decoupling of radial directions, again ? heavy light N=4 Index reduces to a N=1 Index on 3n bosons + 4n fermions 2n bosons + 2n fermions index trivial for a complete intersection in flat ambient space index Kim+Park+P.Y.+Wang 2011 conjectured and computed for many examples Manschot, Pioline, Sen 2010/2011 symmetry and indices of N=4 quantum mechanics basic index of individual BPS particles symmetry and indices of deformed & reduced quantum mechanics symmetry and indices of deformed & reduced quantum mechanics symmetry and indices of deformed & reduced quantum mechanics connecting to the desired index problem Kim+Park+P.Y.+Wang 2011 deformation connecting to the desired index problem Kim+Park+P.Y.+Wang 2011 deformation connecting to the desired index problem Kim+Park+P.Y.+Wang 2011 deformation so far, construction was for a set of distinguishable particles but Bose/Fermi statistics from identical constituent particles is essential, for example, to solve SU(2) U(1) problem. incorporating Bose/Fermi statistics free bulk + sum over fixed submanifolds incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics solution by iteration if there is a universal answer for incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics how ? why ? P.Y. 1997 / Green+Gutperle 1997 incorporating Bose/Fermi statistics final answer with Bose/Fermi statistics incorporated Kim+Park+P.Y.+Wang 2011 Manschot, Pioline, Sen 2010/2011 for example with a pair of states only on the + side which results in, as promised, proved “Coulomb phase” wall-crossing conjecture a la Manschot, Pioline, Sen 2010/2011 from a first principle computation this does not represent the simplest solution to su(2) problem, but it is the most direct derivation of the spectrum from field theory; beyond SU(2), clever mathematical solution are less effective; this approach offers an unique route to computerize the search summary • universal moduli dynamics for BPS dyons/BHs in D=4 N=2 theories • reduction to classical moduli space with reduced supersymmetry • index theorems for
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